Quick Review of Probability

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Quick Review of Probability
Event state of the world that may or may not
occur --- A, B, C, etc. Joint Event events
occurring together --- A and B, AB
BA Conditional Event an event occurring given
that another occurs --- A given B, AB --- B
given A, BA Mut. Exclusive Events events that
cannot occur together Coll. Exhaustive
Events events that together comprise all
possibilities Independent Events events that do
not depend on one another
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P(A) single-event probability P(AB)
P(BA) joint probability P(AB),
P(BA) conditional probability
IMPORTANT FACT If several events are mutually
exclusive and collectively exhaustive, then their
probabilities add to 1.0 The MECE rule
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General Formulas
Conditional Probability P(AB) P(AB) /
P(B) P(BA) P(BA) / P(A) Multiplication
Rule P(AB) P(AB) P(B) P(BA) P(A)
Special Formulas for Independent Events
If you know that A and B are independent
Conditional Probability P(AB) P(A) P(BA)
P(B) Multiplication Rule P(AB) P(A) P(B)
Do not use special formulas unless you know A and
B are independent!
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Probability Trees
A probability tree is a graphical way to
visualize the possible outcomes of a sequence of
occurrences
P(DA)
P(A)
P(EA)
P(FA)
P(B)
P(A) P(B) 1.0
P(DA) P(EA) P(FA) 1.0
P(GB) P(HB) 1.0
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Information on a Probability Tree

• There are two types of probabilities on a
  probability tree
• Single event probabilities
• Conditional probabilities

Joint probabilities are not on the tree, but they
are easy to calculate from the tree. For
example P(AD) P(DA) P(A)
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Tips for Probability Problems

• Always be aware of
• the events (single, joint, or conditional?)
• which probabilities are given
• which probabilities are asked for
• which special situations exist (independence?)
  (may be given, or you may have to deduce)

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Bayes Rule and Too Many Bugs
Public health scientists estimate that 1 of the
general population regularly abuses illegal
substances. As the president of a 2,000-employee
company, you would like to use drug testing to
eliminate this 1 from your work force. A new
drug test has been devised that is relatively
inexpensive and that the manufacturer claims has
a 95 accuracy rate. Your idea is to test each
employee, terminating or keeping the employee
based on the results of the test.
Is this a reasonable and fair plan?
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D event that a person uses Drugs F event that
a person is drug Free A event that the test
Accuses a person E event that the test
Exonerates a person
What does 95 accuracy mean? This is
terminology for how often the test confirms the
truth. P(A D) 0.95 and P(E F) 0.95
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• What are the probabilities of these events?
• Person is accused correctly AD
• Person is accused falsely AF
• Person is exonerated correctly EF
• Person is exonerated falsely ED

• Using the multiplication rule (events not
  independent)
• P(AD) P(A D) P(D) 0.95 0.01 0.0095
• P(AF) P(A F) P(F) 0.05 0.99 0.0495
• P(EF) P(E F) P(F) 0.95 0.99 0.9405
• P(ED) P(E D) P(D) 0.05 0.01 0.0005

Have 4.95 accused falsely, but only 1 actually
use drugs!
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• What are P(A) and P(E)? Because mut. exclusive
• P(A) P(AD) P(AF) 0.0095 0.0495 0.0590
• P(E) P(EF) P(ED) 0.9405 0.0005 0.9410

• What are P(D A) and P(F E)? By the
  conditional probability rule (events not
  independent)
• P(D A) P(AD) / P(A) 0.0095 / 0.0590
  0.1610
• P(F E) P(EF) / P(E) 0.9405 / 0.9410
  0.9995

What does this mean exactly?
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Consider the following situation. Suppose John
Smith has been accused by the drug test. You have
no way of knowing for sure if John uses drugs. Do
you fire John based on the test alone? P( John
uses John has been accused ) P( D A)
0.1610
There is a 16 chance John abuses drugs even
though the test has accused him
Now suppose John Smith has been exonerated by the
test. Do you keep John as an employee? P( John is
free John has been exonerated ) P( F E)
0.9995
Its 99.95 sure that John does not use drugs
based on the result of his test
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The moral of the Bugs article Even if a person
is accused by a test, the conditional
probabilities show that there is a good chance
that the person has been falsely accused.
On the other hand, if a person is exonerated by a
test, then it is nearly certain that the person
deserves to be exonerated.
The process we went through is Bayes Rule, and
it can be used in several additional ways (not
just in drug testing scenarios)
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Test Marketing a New Product
New products introduced in the marketplace have
high sales 8 of the time and low sales 92 of
the time. A marketing test has the following
accuracies if sales are high, then consumer test
reaction is positive 70, neutral 25, and
negative 5 if sales are low, then consumer test
reaction is positive 15, neutral 35, and
negative 50. Your company is developing a new
product and will be test marketing to better
gauge the sales of the new product. Based on
positive, neutral, or negative reactions, what
are the probabilities of high and low sales?
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Decision Analysis
We will now be using probability trees to help us
make decisions in the face of uncertainty

• Ingredients for a quantitative decision
• Possible decisions
• Uncertain events
• Payoffs or costs to decisions and events
• Probabilities of the events

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One day each weekend, you rent an indoor booth to
sell your homemade crafts at the JJ Flea Market.
From your experience, you know the following
Profit/Weather No Rain Rain
Saturday 1000 500
Sunday 700 350
This Saturday, there is a 70 chance of rain, and
this Sunday, there is a 30 chance. Which day do
you sell at the flea market?
Actions sell on Saturday, sell on Sunday Events
rain or not on Saturday, rain or not on Sunday
Payoffs dollar amounts Probs 70, 30, etc.
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How to Evaluate a Decision

• There are two stages to evaluating a decision
• FORWARD stage
• Make a decision tree that lays out all possible
  sequences of decisions and events, along with the
  payoffs and probs
• BACKWARD stage
• Evaluate EMVs or EMCs from the end to the
  beginning

What are EMVs and EMCs? Will explain shortly
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Forward Stage for JJ
0
0
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EMVs and EMCs
EMV Expected Monetary Value the average
payoff of a series of future decisions and
events, assuming that, at any time during the
decision-making process, the decision-maker makes
the wisest decisions given the pending
uncertainties
EMC Expected Monetary Cost the average cost
of a series of future decisions and events,
assuming that, at any time during the
decision-making process, the decision-maker makes
the wisest decisions given the pending
uncertainties
Goal maximize EMV and minimize EMC!
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Backward Stage for JJ
0.7
500
Rain
500
650
0.3
Sat
No Rain
1000
0
1000
650
0.3
350
Rain
350
595
Sun
0.7
0
700
No Rain
700
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Rules for Evaluating EMVs

• The EMVs at the ends of the paths in the tree are
  the total cumulative payoffs gotten over each
  path
• The EMV of an event (circle) is the average of
  the EMVs of the events branches, weighted by the
  probs
• The EMV of a decision (square) is the maximum of
  the EMVs of the decisions branches

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Rules for Evaluating EMCs

• The EMCs at the ends of the paths in the tree are
  the total cumulative costs gotten over each path
• The EMC of an event (circle) is the average of
  the EMCs of the events branches, weighted by the
  probs
• The EMC of a decision (square) is the minimum of
  the EMCs of the decisions branches

EMVs and EMCs are similar except payoffs vs
costs and maximize vs minimize
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Example for EMV of Event
P(A)
EMVA
A
X
EMV
P(B)
EMVB
B
Y
P(C)
EMVC
C
Z
EMV P(A)EMVA P(B)EMVB P(C)EMVC
Note payoffs X, Y, and Z are not in the
formula
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Fruit Flies Discussion(preparation for Homework
2)

• In 1981, Mediterranean fruit flies infested crops
  in Santa Clara County, CA, threatening California
  business
• Spraying for the flies was an option, but
  spraying was environmentally questionable
• Governor Brown and others dismissed spraying
• Instead, a control program of fruit stripping
  was begun, depositing 750 tons of fruit in
  California landfills
• Then, the USDA came up with the sterile male
  solution, attempting to capitalize on a
  biological trait of the flies

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Fruit Flies Discussion(preparation for Homework
2)

• Female fruit flies only mate once
• Radiation was used to sterilize a large quantity
  of male fruit flies
• The males were then released to halt fly
  population growth
• Problem! The irradiation was done incorrectly,
  and the males werent sterile
• More flies, more flies, and more flies
• Embargoes against California fruit
• Governor Brown decides to spray
• But its too late, his career is over